A comparative study of finite frequency scattering rate from Allen, Mitrovi\'c-Fiorucci, Shulga-Dolgov-Maksimov, Sharapov-Carbotte and G\"otze-W\"olfle Memory Function formalisms

We report a comparative study of the scattering rates using different formalisms such as Allen\cite{allen_71}, Shulga et al.\cite{shulga_91}, Mitrovi\'c et al.\cite{mitrovic_85}, Sharapov et al\cite{sharapov_05} and memory function formalism\cite{mori_65}. We discuss the frequency and the temperature dependent scattering rates for the case of electron-phonon interactions in these formalisms. The analysis has been done for different forms of phonon density of states (PDOS) and electron density of states (EDOS). An advantage of our study is that it shed light on the physical assumptions used in these formalisms. From this detailed comparison, we observe that the memory function formalism is the most general one. All other formalisms are based on restrictive assumptions as discussed in this work.